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<span class="kw">let </span><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">B</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font></span><br/>





<span class="lab"><font color="Green" title="E30">A1</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T51" target="_self" title="RELSET_2:th.51">Th51</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E2:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_2"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:65_2"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_2.html#T48" target="_self" title="RELSET_2:th.48">Th48</a></span>;<br/>
<a NAME="E3:65_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T154" title="RELAT_1:th.154">RELAT_1:154</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:65_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E1:65_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E2:65_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:65"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_2.html#K7" title="RELSET_2:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font></span><br/>

<a NAME="E4:65"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">R</font>
 
;<br/>
<a NAME="E5:65"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:65"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
